The Elementary Partial Derivatives

The Elementary Partial Derivatives.—We can set up a number of familiar partial derivatives and thermodynamic formulas, from the information which we already have. We have five variables, of which any two are independent, the rest dependent. We can then set up the partial derivative of any dependent variable with respect to any independent variable, keeping the other independent variable constant. A notation is necessary showing in each case what are the two independent variables. This is a need not ordinarily appreciated in mathematical treatments of partial differentiation, for there the independent variables are usually determined in advance and described in words, so that there is no ambiguity about them. Thus, a notation, peculiar to thermodynamics, has been adopted. In any partial derivative, it is obvious that the quantity being differentiated is one of the dependent variables, and the quantity with respect to which it is differentiated is one of the independent variables. It is only necessary to specify the other independent variable, the one which is held constant in the differentiation, and the convention is to indicate this by a subscript. Thus (dS/dT)P, which is ordinarily read as the partial of S with respect to T at constant P, is the derivative of S in which pressure and temperature are independent variables. This derivative would mean an entirely different thing from the derivative of S with respect to T at constant V, for instance. 

There are a number of partial derivatives which have elementary meanings. Thus, consider the thermal expansion. This is the fractional increase of volume per unit rise of temperature, at constant pressure  

Similarly, the isothermal compressibility is the fractional decrease of volume per unit increase of pressure, at constant temperature  

This is the compressibility usually employed sometimes, as in considering sound waves, we require the adiabatic compressibility, the fractional decrease of volume per unit increase of pressure, when no heat flows in or out. If there is no heat flow, the entropy is unchanged, in a reversible process, so that an adiabatic process is one at constant entropy. Then we have 

The specific heats have simple formulas. At constant volume, the heat absorbed equals the increase of internal energy, since no work is done. Since the heat absorbed also equals the temperature times the change of entropy, for a reversible process, and since the heat capacity at constant volume Cv is the heat absorbed per unit change of temperature at constant volume, we have the alternative formulas 

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